Abstract
We study a variation of Bagchi and Datta's $\sigma$-vector of a simplicial complex $C$, whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of $C$. We show that these invariants satisfy an Alexander-Dehn-Sommerville type identity, and behave nicely under natural operations on triangulated manifolds and spheres such as connected sums and bistellar flips. In the language of commutative algebra, the invariants are weighted sums of graded Betti numbers of the Stanley-Reisner ring of $C$. This interpretation implies, by a result of Adiprasito, that the Billera-Lee sphere maximizes these invariants among triangulated spheres with a given $f$-vector. For the first entry of $\sigma$, we extend this bound to the class of strongly connected pure complexes. As an application, we show how upper bounds on $\sigma$ can be used to obtain lower bounds on the $f$-vector of triangulated $4$-manifolds with transitive symmetry on vertices and prescribed vector of Betti numbers.
Highlights
In this article we investigate a combinatorial invariant of simplicial complexes that we call the τ -vector, defined as follows: for a simplicial complex C with ground set V, and for each i = {−1, 0, 1, 2, . . . }, 1 τi(C) = |V | + 1 βi(C[W ]) . |V |
We present an example of a tight triangulation of a 4-manifold whose vertex links do not have maximal τ -vectors amongst all 3-spheres with a given f -vector
A combinatorial manifold is a simplicial complex in which the link of every vertex v, i.e., the boundary of the complex consisting of all facets of C containing v and their faces, is a triangulated sphere PL-homeomorphic to the standard sphere
Summary
The τ -vector is interesting from a more combinatorial viewpoint It produces the following characterization of stacked spheres among normal pseudo-manifolds (see Section 5.2 for more details): Theorem 1.2 (Murai [41, Corollary 5.8.(ii)] for the general case, Burton, Datta, Singh and Spreer [16, Theorem 1.1] for 2-spheres). We present an example of a tight triangulation of a 4-manifold whose vertex links do not have maximal τ -vectors amongst all 3-spheres with a given f -vector (see Theorem 6.8) This is surprising in light of the Lutz-Kuhnel conjecture, see Theorem 6.4 and [33]
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